Discussion

Find the value of \(\left(\dfrac{81}{16}\right)^{-3/4}\).

Answer: B

Correct answer: 8/27

Recognise the base as a fourth power.

\[\frac{81}{16} = \left(\frac{3}{2}\right)^4\]

\[\left(\left(\frac{3}{2}\right)^4\right)^{-3/4} = \left(\frac{3}{2}\right)^{-3} = \left(\frac{2}{3}\right)^3 = \frac{8}{27}\]

The negative sign in the index inverts the fraction; the denominator 4 takes the fourth root and the numerator 3 cubes the result.

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